On the Convergence of Convolved Vector Subdivision Schemes

The aim of this paper is to discuss the use of convolution to build new families of refinable functions and thus new families of subdivision algorithms. In particular, we first generalise a result given in [C. Conti, K. Jetter, in: A. Cohen, R. Rabut, L.L. Schumaker (Eds.), Curves and Surfaces Fitting, Saint Malo, 1999, Vanderbilt University Press, Nashville, TN, 2000, pp. 135–142] where, following a too restrictive definition of convergence, it is proved that convolution of two refinable function vectors, which give rise to convergent subdivision schemes, generates a convolved scheme which is again convergent. Then, we discuss the use of convolution to construct refinable bases suitable to interpolate directions given by difference of control points. This turns out to be useful, for example, in algebraic numerical grid generation methods such as the one discussed in [C. Conti, R. Morandi, in: Proceedings of MASCOT/02, IMACS Ser. Comput. Appl. Math., IMACS, New Brunswick, NJ, 2003, pp. 75–82].